Problem 7.26 — For each rational homology 3-sphere M, calculate τ SO(3)(M ) and τ P SU (N )(M ) for all degrees.

Definitions and mathematical statement

A rational homology three-sphere is a closed three-manifold $M$ with $H_i(M;\mathbb Q)\cong H_i(S^3;\mathbb Q)$; the quantum-invariant context also requires an orientation convention. Let $\operatorname{SO}(3)$ be the rotation group and $\operatorname{PSU}(N)=\operatorname{SU}(N)/Z(\operatorname{SU}(N))$, $N\ge2$.

For every rational homology three-sphere, the source asks to calculate \[ \tau^{\operatorname{SO}(3)}(M)\quad\text{and}\quad \tau^{\operatorname{PSU}(N)}(M) \] "for all degrees". The precise $\tau$-invariants are not defined in the excerpt. If they are formal expansions $\tau^G(M)=\sum_{d\ge0}\tau_d^G(M)h^d$, the requested output is every coefficient $\tau_d^G(M)$; this is only a conditional interpretation, not a chosen expansion parameter. Degrees of a perturbative series and root-of-unity quantum levels are different indices and are not silently equated.

The normalization, coefficient ring, expansion point or level restrictions, auxiliary data and intended computational form are absent. A formula, recurrence, effective coefficient algorithm or tables are different possible meanings of calculate. The rational-homology-sphere scope is retained rather than restricted to integral homology spheres. The summary supplies no exact result beyond its open/triage label, which remains source information pending current primary-literature review.

Short English statement

Compute the specified $\operatorname{SO}(3)$ and $\operatorname{PSU}(N)$ invariants for every rational homology sphere in every intended degree. The invariant definitions and meaning of degree are missing, so levels and perturbative coefficients must not be confused.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10400141$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 7.26, PDF page 117: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Supplied PDF-font-structure extraction accessed 2026-07-29; provenance distinguishes 231 numbered items.
• No preceding invariant definitions, exact computations or separate current resolution reference are supplied.